Abstract

We experimentally and numerically study nonlinear light propagation in a fractal waveguide array. We consider a nested set of periodic arrays and examine energy transport as a function of band structure, nonlinearity, and probe beam geometry. Experimentally, we observe the behavior directly by recording intensity in position space and power spectra in momentum space. The results are fundamental to nonlinear wave dynamics in self-similar structures and hold potential to improve the efficiency and sensitivity of fractal photonic devices.

© 2010 OSA

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  1. R. W. Boyd, Nonlinear Optics, (Academic Press, 2008).
  2. D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100(10), 103903 (2008).
    [CrossRef] [PubMed]
  3. D. V. Dylov and J. W. Fleischer, “Multiple-stream instabilities and soliton turbulence in photonic plasma,” Phys. Rev. A 78(6), 061804 (2008).
    [CrossRef]
  4. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
    [CrossRef] [PubMed]
  5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
    [CrossRef]
  6. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
    [CrossRef] [PubMed]
  7. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
    [CrossRef] [PubMed]
  8. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
    [CrossRef] [PubMed]
  9. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
    [CrossRef] [PubMed]
  10. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
    [CrossRef]
  11. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
    [CrossRef] [PubMed]
  12. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
    [CrossRef] [PubMed]
  13. P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
    [CrossRef] [PubMed]
  14. B. B. Mandelbrot, The Fractal Geometry of Nature, (W. H. Freeman and Company, 1982).
  15. M. Soljacic, M. Segev, and C. R. Menyuk, “Self similarity and fractals in soliton-supporting systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 1048 (2000).
    [CrossRef]
  16. S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
    [CrossRef] [PubMed]
  17. G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
    [CrossRef]
  18. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
    [CrossRef]
  19. C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
    [CrossRef] [PubMed]
  20. F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14(25), 11958–11963 (2006).
    [CrossRef] [PubMed]
  21. D. H. Werner and P. L. Werner, “On the synthesis of fractal radiation patterns,” Radio Sci. 30(1), 29–45 (1995).
    [CrossRef]
  22. N. Cohen, “Fractal Antennas,” Communications Quarterly, Summer, 1995
  23. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
    [CrossRef] [PubMed]
  24. N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
    [CrossRef]
  25. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
    [CrossRef] [PubMed]
  26. M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot Fractal Function,” Proc. R. Soc. Lond. A Math. Phys. Sci. 370(1743), 459–484 (1980).
    [CrossRef]
  27. J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. 85(25), 5320–5323 (2000).
    [CrossRef]
  28. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177(1-6), 9–13 (2000).
    [CrossRef]
  29. S. Blair, “Nonparaxial one-dimensional spatial solitons,” Chaos 10(3), 570–583 (2000).
    [CrossRef]
  30. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
    [CrossRef] [PubMed]
  31. C. Tricot, Curves and fractal dimension, (Springer-Verlag, New York, 1995).
  32. S. Jia, W. Wan, and J. W. Fleischer, “Dispersive shock waves in nonlinear arrays,” Phys. Rev. Lett. 99(22), 223901 (2007).
    [CrossRef]
  33. J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94(17), 174101 (2005).
    [CrossRef] [PubMed]

2008

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100(10), 103903 (2008).
[CrossRef] [PubMed]

D. V. Dylov and J. W. Fleischer, “Multiple-stream instabilities and soliton turbulence in photonic plasma,” Phys. Rev. A 78(6), 061804 (2008).
[CrossRef]

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

2007

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

S. Jia, W. Wan, and J. W. Fleischer, “Dispersive shock waves in nonlinear arrays,” Phys. Rev. Lett. 99(22), 223901 (2007).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

2006

F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14(25), 11958–11963 (2006).
[CrossRef] [PubMed]

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
[CrossRef] [PubMed]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

2005

2004

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

2003

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

2002

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

2001

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

2000

M. Soljacic, M. Segev, and C. R. Menyuk, “Self similarity and fractals in soliton-supporting systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 1048 (2000).
[CrossRef]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. 85(25), 5320–5323 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177(1-6), 9–13 (2000).
[CrossRef]

S. Blair, “Nonparaxial one-dimensional spatial solitons,” Chaos 10(3), 570–583 (2000).
[CrossRef]

1999

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
[CrossRef]

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

1995

D. H. Werner and P. L. Werner, “On the synthesis of fractal radiation patterns,” Radio Sci. 30(1), 29–45 (1995).
[CrossRef]

1980

M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot Fractal Function,” Proc. R. Soc. Lond. A Math. Phys. Sci. 370(1743), 459–484 (1980).
[CrossRef]

1979

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Aitchison, J. S.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[CrossRef] [PubMed]

Barthelemy, P.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[CrossRef] [PubMed]

Bergman, K.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot Fractal Function,” Proc. R. Soc. Lond. A Math. Phys. Sci. 370(1743), 459–484 (1980).
[CrossRef]

Bertolotti, J.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[CrossRef] [PubMed]

Blair, S.

S. Blair, “Nonparaxial one-dimensional spatial solitons,” Chaos 10(3), 570–583 (2000).
[CrossRef]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Buljan, H.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177(1-6), 9–13 (2000).
[CrossRef]

Cohen, O.

Conti, C.

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
[CrossRef] [PubMed]

Courtial, J.

J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. 85(25), 5320–5323 (2000).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177(1-6), 9–13 (2000).
[CrossRef]

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177(1-6), 9–13 (2000).
[CrossRef]

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[CrossRef]

Dylov, D. V.

D. V. Dylov and J. W. Fleischer, “Multiple-stream instabilities and soliton turbulence in photonic plasma,” Phys. Rev. A 78(6), 061804 (2008).
[CrossRef]

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100(10), 103903 (2008).
[CrossRef] [PubMed]

Efremidis, N.

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Finot, C.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Fleischer, J. W.

D. V. Dylov and J. W. Fleischer, “Multiple-stream instabilities and soliton turbulence in photonic plasma,” Phys. Rev. A 78(6), 061804 (2008).
[CrossRef]

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100(10), 103903 (2008).
[CrossRef] [PubMed]

S. Jia, W. Wan, and J. W. Fleischer, “Dispersive shock waves in nonlinear arrays,” Phys. Rev. Lett. 99(22), 223901 (2007).
[CrossRef]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

Freedman, B.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[CrossRef] [PubMed]

Furlan, W. D.

Genov, D. A.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Ghofraniha, N.

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
[CrossRef] [PubMed]

Giménez, F.

Huang, J. G.

J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94(17), 174101 (2005).
[CrossRef] [PubMed]

Jia, S.

S. Jia, W. Wan, and J. W. Fleischer, “Dispersive shock waves in nonlinear arrays,” Phys. Rev. Lett. 99(22), 223901 (2007).
[CrossRef]

Karman, G. P.

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
[CrossRef]

Krylov, D.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

Kukhtarev, N.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

Lewis, Z. V.

M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot Fractal Function,” Proc. R. Soc. Lond. A Math. Phys. Sci. 370(1743), 459–484 (1980).
[CrossRef]

Lifshitz, R.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Manela, O.

Markov, V. B.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

McDonald, G. S.

J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94(17), 174101 (2005).
[CrossRef] [PubMed]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
[CrossRef]

Meier, J.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

Menyuk, C. R.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self similarity and fractals in soliton-supporting systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 1048 (2000).
[CrossRef]

Millot, G.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[CrossRef]

Monsoriu, J. A.

Morandotti, R.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

New, G. H. C.

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
[CrossRef]

Odulov, S. G.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Padgett, M. J.

J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. 85(25), 5320–5323 (2000).
[CrossRef]

Pons, A.

Richardson, D. J.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[CrossRef]

Ruocco, G.

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
[CrossRef] [PubMed]

Salamo, G.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[CrossRef] [PubMed]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, and C. R. Menyuk, “Self similarity and fractals in soliton-supporting systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 1048 (2000).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Soljacic, M.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, and C. R. Menyuk, “Self similarity and fractals in soliton-supporting systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 1048 (2000).
[CrossRef]

Sorel, M.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

Soskin, M. S.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

Trillo, S.

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
[CrossRef] [PubMed]

Vinetskii, V. L.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Wan, W.

S. Jia, W. Wan, and J. W. Fleischer, “Dispersive shock waves in nonlinear arrays,” Phys. Rev. Lett. 99(22), 223901 (2007).
[CrossRef]

Werner, D. H.

D. H. Werner and P. L. Werner, “On the synthesis of fractal radiation patterns,” Radio Sci. 30(1), 29–45 (1995).
[CrossRef]

Werner, P. L.

D. H. Werner and P. L. Werner, “On the synthesis of fractal radiation patterns,” Radio Sci. 30(1), 29–45 (1995).
[CrossRef]

Wiersma, D. S.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[CrossRef] [PubMed]

Woerdman, J. P.

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
[CrossRef]

Yang, H.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

Zhang, X.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Chaos

S. Blair, “Nonparaxial one-dimensional spatial solitons,” Chaos 10(3), 570–583 (2000).
[CrossRef]

Ferroelectrics

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979).
[CrossRef]

Nat. Phys.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[CrossRef]

Nature

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, “Laser optics: Fractal modes in unstable resonators,” Nature 402(6758), 138 (1999).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[CrossRef] [PubMed]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

Opt. Commun.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177(1-6), 9–13 (2000).
[CrossRef]

Opt. Express

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete Solitons in Optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[CrossRef]

Phys. Rev. A

D. V. Dylov and J. W. Fleischer, “Multiple-stream instabilities and soliton turbulence in photonic plasma,” Phys. Rev. A 78(6), 061804 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046602 (2002).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

M. Soljacic, M. Segev, and C. R. Menyuk, “Self similarity and fractals in soliton-supporting systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 1048 (2000).
[CrossRef]

Phys. Rev. Lett.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84(9), 1902–1905 (2000).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[CrossRef] [PubMed]

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100(10), 103903 (2008).
[CrossRef] [PubMed]

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser beam filamentation in fractal aggregates,” Phys. Rev. Lett. 97(12), 123903 (2006).
[CrossRef] [PubMed]

J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. 85(25), 5320–5323 (2000).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

S. Jia, W. Wan, and J. W. Fleischer, “Dispersive shock waves in nonlinear arrays,” Phys. Rev. Lett. 99(22), 223901 (2007).
[CrossRef]

J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94(17), 174101 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci.

M. V. Berry and Z. V. Lewis, “On the Weierstrass-Mandelbrot Fractal Function,” Proc. R. Soc. Lond. A Math. Phys. Sci. 370(1743), 459–484 (1980).
[CrossRef]

Radio Sci.

D. H. Werner and P. L. Werner, “On the synthesis of fractal radiation patterns,” Radio Sci. 30(1), 29–45 (1995).
[CrossRef]

Other

N. Cohen, “Fractal Antennas,” Communications Quarterly, Summer, 1995

C. Tricot, Curves and fractal dimension, (Springer-Verlag, New York, 1995).

R. W. Boyd, Nonlinear Optics, (Academic Press, 2008).

B. B. Mandelbrot, The Fractal Geometry of Nature, (W. H. Freeman and Company, 1982).

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Figures (6)

Fig. 1
Fig. 1

Setup and properties of the fractal waveguide array. (a) Experimental setup. Light from a 532 nm laser is first split by a polarizing beam splitter. The ordinarily-polarized beam is phase-modulated into a fractal lattice by a spatial light modulator (SLM). The extraordinarily-polarized beam is used as a probe, e.g. as a focused Gaussian beam. The two beams then are recombined onto an SBN:75 photorefractive crystal. Light exiting the crystal is imaged into two CCD cameras, one for the intensity in position (x) space, one for the power spectrum in momentum (k) space (obtained by performing an optical Fourier transform). (b) Experimental and numerical pictures of the fractal waveguide array and the experimental Fourier structure of the array.

Fig. 2
Fig. 2

Band structures of single-, double-, and triple-period lattices. The dashed lines in the middle picture show the process of band folding.

Fig. 3
Fig. 3

Experimental output spectra (k-space intensity) of a plane wave probe beam incident into lattice of (a, b) single period of 28μm, (c, d) superposition of periods of 14μm, 28μm and 56μm, and (e, f) superposition of periods of 7μm, 14μm, 28μm, 56μm and 112μm. (a, c, e) are identical, showing the full five-period lattice, while the arrows indicate the periods actually used in (b, d, f), respectively. Each arrow can be observed to be shifted by twice the smallest scales used. The inset in (a) is the input (plane wave at k = 0, dash lines in (a, c, e)).

Fig. 5
Fig. 5

Output intensities of a Gaussian probe beam incident into the central site of the fractal lattice. Top row: numerical simulation; bottom row: experiment. Shown are linear (a, b) and nonlinear (c,d) propagation.

Fig. 4
Fig. 4

Spectral results of a fractal probe beam. (a) Profile of probe beam [red] incident into the fractal lattice [blue]. (b) Theoretical analysis shows that each initial mode (solid line) is shifted by twice the smallest scale to new modes (dashed line). (c) Input and (d) nonlinear output. Each arrow can be observed to be shifted by twice the smallest scales used.

Fig. 6
Fig. 6

Numerical study showing the relation of fractal dimension vs. nonlinearity. (a) Different locations of the Gaussian probe beam. (b) Fractal dimension of the output, as measured by box-counting.

Equations (1)

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i ψ z + 1 2 k 0 2 ψ + V ( x ) ψ + Δ n ( | ψ | 2 ) ψ = 0

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